MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  opnnei Structured version   Visualization version   GIF version

Theorem opnnei 23143
Description: A set is open iff it is a neighborhood of all of its points. (Contributed by Jeff Hankins, 15-Sep-2009.)
Assertion
Ref Expression
opnnei (𝐽 ∈ Top → (𝑆𝐽 ↔ ∀𝑥𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥})))
Distinct variable groups:   𝑥,𝐽   𝑥,𝑆

Proof of Theorem opnnei
StepHypRef Expression
1 0opn 22925 . . . . 5 (𝐽 ∈ Top → ∅ ∈ 𝐽)
21adantr 480 . . . 4 ((𝐽 ∈ Top ∧ 𝑆 = ∅) → ∅ ∈ 𝐽)
3 eleq1 2826 . . . . 5 (𝑆 = ∅ → (𝑆𝐽 ↔ ∅ ∈ 𝐽))
43adantl 481 . . . 4 ((𝐽 ∈ Top ∧ 𝑆 = ∅) → (𝑆𝐽 ↔ ∅ ∈ 𝐽))
52, 4mpbird 257 . . 3 ((𝐽 ∈ Top ∧ 𝑆 = ∅) → 𝑆𝐽)
6 rzal 4514 . . . 4 (𝑆 = ∅ → ∀𝑥𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥}))
76adantl 481 . . 3 ((𝐽 ∈ Top ∧ 𝑆 = ∅) → ∀𝑥𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥}))
85, 72thd 265 . 2 ((𝐽 ∈ Top ∧ 𝑆 = ∅) → (𝑆𝐽 ↔ ∀𝑥𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥})))
9 opnneip 23142 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑆𝐽𝑥𝑆) → 𝑆 ∈ ((nei‘𝐽)‘{𝑥}))
1093expia 1120 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑆𝐽) → (𝑥𝑆𝑆 ∈ ((nei‘𝐽)‘{𝑥})))
1110ralrimiv 3142 . . . . 5 ((𝐽 ∈ Top ∧ 𝑆𝐽) → ∀𝑥𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥}))
1211ex 412 . . . 4 (𝐽 ∈ Top → (𝑆𝐽 → ∀𝑥𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥})))
1312adantr 480 . . 3 ((𝐽 ∈ Top ∧ ¬ 𝑆 = ∅) → (𝑆𝐽 → ∀𝑥𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥})))
14 df-ne 2938 . . . . . 6 (𝑆 ≠ ∅ ↔ ¬ 𝑆 = ∅)
15 r19.2z 4500 . . . . . . 7 ((𝑆 ≠ ∅ ∧ ∀𝑥𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥})) → ∃𝑥𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥}))
1615ex 412 . . . . . 6 (𝑆 ≠ ∅ → (∀𝑥𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥}) → ∃𝑥𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥})))
1714, 16sylbir 235 . . . . 5 𝑆 = ∅ → (∀𝑥𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥}) → ∃𝑥𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥})))
18 eqid 2734 . . . . . . . 8 𝐽 = 𝐽
1918neii1 23129 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑆 ∈ ((nei‘𝐽)‘{𝑥})) → 𝑆 𝐽)
2019ex 412 . . . . . 6 (𝐽 ∈ Top → (𝑆 ∈ ((nei‘𝐽)‘{𝑥}) → 𝑆 𝐽))
2120rexlimdvw 3157 . . . . 5 (𝐽 ∈ Top → (∃𝑥𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥}) → 𝑆 𝐽))
2217, 21sylan9r 508 . . . 4 ((𝐽 ∈ Top ∧ ¬ 𝑆 = ∅) → (∀𝑥𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥}) → 𝑆 𝐽))
2318ntrss2 23080 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ 𝑆 𝐽) → ((int‘𝐽)‘𝑆) ⊆ 𝑆)
2423adantr 480 . . . . . . . . . 10 (((𝐽 ∈ Top ∧ 𝑆 𝐽) ∧ ∀𝑥𝑆 {𝑥} ⊆ ((int‘𝐽)‘𝑆)) → ((int‘𝐽)‘𝑆) ⊆ 𝑆)
25 vex 3481 . . . . . . . . . . . . 13 𝑥 ∈ V
2625snss 4789 . . . . . . . . . . . 12 (𝑥 ∈ ((int‘𝐽)‘𝑆) ↔ {𝑥} ⊆ ((int‘𝐽)‘𝑆))
2726ralbii 3090 . . . . . . . . . . 11 (∀𝑥𝑆 𝑥 ∈ ((int‘𝐽)‘𝑆) ↔ ∀𝑥𝑆 {𝑥} ⊆ ((int‘𝐽)‘𝑆))
28 dfss3 3983 . . . . . . . . . . . . 13 (𝑆 ⊆ ((int‘𝐽)‘𝑆) ↔ ∀𝑥𝑆 𝑥 ∈ ((int‘𝐽)‘𝑆))
2928biimpri 228 . . . . . . . . . . . 12 (∀𝑥𝑆 𝑥 ∈ ((int‘𝐽)‘𝑆) → 𝑆 ⊆ ((int‘𝐽)‘𝑆))
3029adantl 481 . . . . . . . . . . 11 (((𝐽 ∈ Top ∧ 𝑆 𝐽) ∧ ∀𝑥𝑆 𝑥 ∈ ((int‘𝐽)‘𝑆)) → 𝑆 ⊆ ((int‘𝐽)‘𝑆))
3127, 30sylan2br 595 . . . . . . . . . 10 (((𝐽 ∈ Top ∧ 𝑆 𝐽) ∧ ∀𝑥𝑆 {𝑥} ⊆ ((int‘𝐽)‘𝑆)) → 𝑆 ⊆ ((int‘𝐽)‘𝑆))
3224, 31eqssd 4012 . . . . . . . . 9 (((𝐽 ∈ Top ∧ 𝑆 𝐽) ∧ ∀𝑥𝑆 {𝑥} ⊆ ((int‘𝐽)‘𝑆)) → ((int‘𝐽)‘𝑆) = 𝑆)
3332ex 412 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑆 𝐽) → (∀𝑥𝑆 {𝑥} ⊆ ((int‘𝐽)‘𝑆) → ((int‘𝐽)‘𝑆) = 𝑆))
3425snss 4789 . . . . . . . . . . . 12 (𝑥𝑆 ↔ {𝑥} ⊆ 𝑆)
35 sstr2 4001 . . . . . . . . . . . . . 14 ({𝑥} ⊆ 𝑆 → (𝑆 𝐽 → {𝑥} ⊆ 𝐽))
3635com12 32 . . . . . . . . . . . . 13 (𝑆 𝐽 → ({𝑥} ⊆ 𝑆 → {𝑥} ⊆ 𝐽))
3736adantl 481 . . . . . . . . . . . 12 ((𝐽 ∈ Top ∧ 𝑆 𝐽) → ({𝑥} ⊆ 𝑆 → {𝑥} ⊆ 𝐽))
3834, 37biimtrid 242 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ 𝑆 𝐽) → (𝑥𝑆 → {𝑥} ⊆ 𝐽))
3938imp 406 . . . . . . . . . 10 (((𝐽 ∈ Top ∧ 𝑆 𝐽) ∧ 𝑥𝑆) → {𝑥} ⊆ 𝐽)
4018neiint 23127 . . . . . . . . . . . 12 ((𝐽 ∈ Top ∧ {𝑥} ⊆ 𝐽𝑆 𝐽) → (𝑆 ∈ ((nei‘𝐽)‘{𝑥}) ↔ {𝑥} ⊆ ((int‘𝐽)‘𝑆)))
41403com23 1125 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ 𝑆 𝐽 ∧ {𝑥} ⊆ 𝐽) → (𝑆 ∈ ((nei‘𝐽)‘{𝑥}) ↔ {𝑥} ⊆ ((int‘𝐽)‘𝑆)))
42413expa 1117 . . . . . . . . . 10 (((𝐽 ∈ Top ∧ 𝑆 𝐽) ∧ {𝑥} ⊆ 𝐽) → (𝑆 ∈ ((nei‘𝐽)‘{𝑥}) ↔ {𝑥} ⊆ ((int‘𝐽)‘𝑆)))
4339, 42syldan 591 . . . . . . . . 9 (((𝐽 ∈ Top ∧ 𝑆 𝐽) ∧ 𝑥𝑆) → (𝑆 ∈ ((nei‘𝐽)‘{𝑥}) ↔ {𝑥} ⊆ ((int‘𝐽)‘𝑆)))
4443ralbidva 3173 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑆 𝐽) → (∀𝑥𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥}) ↔ ∀𝑥𝑆 {𝑥} ⊆ ((int‘𝐽)‘𝑆)))
4518isopn3 23089 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑆 𝐽) → (𝑆𝐽 ↔ ((int‘𝐽)‘𝑆) = 𝑆))
4633, 44, 453imtr4d 294 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑆 𝐽) → (∀𝑥𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥}) → 𝑆𝐽))
4746ex 412 . . . . . 6 (𝐽 ∈ Top → (𝑆 𝐽 → (∀𝑥𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥}) → 𝑆𝐽)))
4847com23 86 . . . . 5 (𝐽 ∈ Top → (∀𝑥𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥}) → (𝑆 𝐽𝑆𝐽)))
4948adantr 480 . . . 4 ((𝐽 ∈ Top ∧ ¬ 𝑆 = ∅) → (∀𝑥𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥}) → (𝑆 𝐽𝑆𝐽)))
5022, 49mpdd 43 . . 3 ((𝐽 ∈ Top ∧ ¬ 𝑆 = ∅) → (∀𝑥𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥}) → 𝑆𝐽))
5113, 50impbid 212 . 2 ((𝐽 ∈ Top ∧ ¬ 𝑆 = ∅) → (𝑆𝐽 ↔ ∀𝑥𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥})))
528, 51pm2.61dan 813 1 (𝐽 ∈ Top → (𝑆𝐽 ↔ ∀𝑥𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥})))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395   = wceq 1536  wcel 2105  wne 2937  wral 3058  wrex 3067  wss 3962  c0 4338  {csn 4630   cuni 4911  cfv 6562  Topctop 22914  intcnt 23040  neicnei 23120
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-top 22915  df-ntr 23043  df-nei 23121
This theorem is referenced by:  neiptopreu  23156  flimcf  24005
  Copyright terms: Public domain W3C validator